178 XIII INTEGRATION linear isometry Up of L£(X, JJL) onto L£(Y, v) which extends £/. (Show that U has a unique extension to a linear bijection of the space E(X, p) of classes of /x-integrable step functions on X, onto the space E(Y, v) of classes of v-integrable step functions on Y. Begin by proving that if inf(A, S) = $9 then also inf(£/(A), C/(B)) = <f> and Z7(sup(A, S)) = sup(tf(A), t/(B)); then deduce that U(A) <i t/(B) whenever A <^8; finally, show that for arbitrary A, S we have £/(inf(A, S)) = inf(C/(A), t/(S)) and t/(sup(A, B)) = sup(t/(A), tf(8)). Observe then that Np(<7(/)) = Np(/) for/e E(X5 /*).) Show that the image of Lg(X, jit) n L£(X, ^) under C/p is LJ(Y, v) n L£(Y, v),and that the restriction of Up to L£(X, ju,) n L^X, ft) is an isomorphism of Banach algebras. (d) Conversely, for a p such that 1 ^p< -foo, let V be a linear isometry of L£(X, jit) onto L£(Y, v) such that the restriction of V to LJ(X, fi) n L£(X, ^) is an algebra isomorphism onto Lg(Y, v) n Lg(Y, v). Then the restriction of Fto I(X, p,) is an isometry onto I(Y, v) which maps f to $. 9. Let X be a compact space such that //,(X) = 1. Let % be a set of integrable subsets of X, such that the relations A e % and B e % imply that X — A e X and A n B e %. Suppose also that the set {A: A e £} is dense in the metric space I(X, fji) (Problem 8). (a) Let (Cj)i^j^« be a finite partition of X into /x-integrable subsets. Show that for each €>0 there exists a partition (A^i^j^n consisting of sets in X such that /x(D(Q, A,)) <; e for 1 <,j <£ n. (Observe that if ju,(D(C,, B.,)) <j 8 for 1 gy ^ n - 1, then jU,(Bj n Bk) < 28 for 1 <;/ < k :g « — 1; if N is the union of the By n Bfc, consider the sets A, = B, -N (1 ^/ <i TZ - 1) and An = X -"Q A,). (b) Deduce from (a) that for each finite partition y = (Q)i <J<n of X into integrable subsets, and each e>0, there exists a finite partition a==(Aj)i^j^n consisting of sets in % and such that (in the notation of Section 13.9, Problem 27) H(a/y) + H(y/a) ^ fi. (Reduce to the case where none of the Cj is negligible, and use the fact that the function r«-> t log r is zero and continuous at t = 1.) 10. Let X be a compact space such that ju,(X) = 1, and let u; X-»X be a /^-measurable mapping such that u(^) — jit. (a) Let a be a finite partition of X into integrable subsets, and let % be the set of n- 1 all finite unions of subsets of X belonging to one or other of the partitions V u~J(oc). J = 0 Suppose that the set of classes {A: A e X} is dense in I(X, ft) (Problem 8). Show that we then have h(u) = h(u, a). (It is enough to show that h(u, /?) <[ h(u, a) for every finite partition J3 of X into integrable subsets; observe that we have V u-J(oc)\ +H(B/ \) u-J(o J = o ] \ I J~o and use Problem 28(d) of Section 13.9, and Problem 9 above.) (b) Under the hypotheses of (a), show that if in addition u is bijective and u""1 is jit-measurable, then h(u) — 0. (Observe that the set of classes {(zrXA))"; A e%] is again dense in I(X, ft), and use Problem 28(c) of Section 13.9.) (c) Suppose that u is bijective and u~~l is /x-measurable. Let £' be the set of finite unions of subsets of X belonging to one or other of the partitions V «J(a), and supposee the graph of w, and